ICTP–WE Heraeus College · Advances in Topological Quantum Matter

Absence of quasi-Majorana false positives in full-shell hybrid nanowires

Carlos Payá

Instituto de Ciencia de Materiales de Madrid (ICMM), CSIC

Trieste · 30 October 2026

Phys. Rev. Lett. 137, 096606 (2026)

A zero-bias peak is the main signature of a Majorana.
It is also the easiest one to fake.

Can a smooth end still fake it
in a full-shell nanowire?

Quasi-Majorana false positives

A nanowire turns topological above a critical Zeeman field

H = \Bigl[\tfrac{p_z^2}{2m^*} - \mu\Bigr]\tau_z + \alpha\, p_z \sigma_y \tau_z + V_\mathrm{Z}\,\sigma_z + \Delta\,\tau_x

Gap closes at V_\mathrm{Z}^\mathrm{c} = \sqrt{\Delta^2 + \mu^2}

Reopens inverted (Q = -1): a Majorana (MZM) at each end

Class D: particle–hole symmetry alone, shared by all subbands

Strictly BDI in this ideal 1D model; disorder makes it D

R. M. Lutchyn et al. PRL 105, 077001 (2010) · Y. Oreg et al. PRL 105, 177002 (2010) · A. Kitaev, Phys. Usp. 44, 131 (2001)

(1:20) The Majorana nanowire you have seen this week: a Rashba semiconductor core with a superconductor on one side, the partial shell, under a field along the wire, probed at its end (paper’s partial-shell parameters, constant Δ here). [click] As V_Z grows, the gap at k = 0 closes at V_Z^c = √(Δ² + μ²). [click] Beyond it, it reopens inverted: the Pfaffian index flips sign and a Majorana sits at each end. The inset is the phase diagram, the dot is our wire at μ_bulk. [click] I want to insist on the symmetry: all we need is particle–hole symmetry, which is built into the BdG formalism, so no disorder can remove it: class D, a single Hamiltonian that all subbands share. Keep this in mind: it is what changes in full shells. The small print: this ideal 1D Hamiltonian is real, an effective time reversal with square +1, so strictly it is BDI with an integer index. Disorder, several subbands or a field along the spin–orbit axis break it; only the parity survives: class D. Nothing in this talk relies on it. Drag the slider for questions. (If asked: −iσ_y is real, so α p_z σ_y is real; 𝒯 = K. Physically, T times a mirror.)

A Majorana shows up as a zero-bias peak at the end of the wire

A normal probe tunnels into the end: dI/dV \propto local DOS at z = 0
Abrupt end: the bulk gap closes at V_\mathrm{Z}^\mathrm{c} and a zero-bias peak (ZBP) follows

A ZBP is necessary but not sufficient: trivial states can also stick to zero energy

V. Mourik et al. Science 336, 1003 (2012) · E. Prada et al. Nat. Rev. Phys. 2, 575 (2020)

A smooth end makes the wire topological only locally

The tunnel gate depletes the end; the potential heals over a length \chi: \;\mu(z) = \mu_\mathrm{bulk}\bigl(1 - e^{-z/\chi}\bigr), topological wherever V_\mathrm{Z} > \sqrt{\Delta^2 + \mu(z)^2}

E. Prada et al. PRB 86, 180503(R) (2012) · G. Kells et al. PRB 86, 100503(R) (2012) · C. Payá et al. PRL 137, 096606 (2026)

Smooth ends pin the topological section’s two Majoranas to zero

\Psi_\mathrm{L} and \Psi_\mathrm{R} still overlap, but a smooth end cannot couple their different oscillations

G. Kells et al. PRB 86, 100503(R) (2012) · A. Vuik et al. SciPost Phys. 7, 061 (2019) · F. Peñaranda et al. PRB 98, 235406 (2018)

(1:30) Here is that pair for the paper’s wire at V_Z^(2), for many χ. Left, the two Majoranas themselves, signed, over the shaded region where each lives. Ψ_L, at the hard end, oscillates with the Fermi wavelength, about 120 nm. Ψ_R, at the soft edge of the topological section, is a smooth bump. At χ = 50 nm they overlap strongly and the pair is split (bottom right, black). [click] Now make the end smoother. Watch the shaded regions: they keep overlapping. The overlap (dashed) drops by less than a decade, while the splitting falls by four: below 10⁻² Δ0 at χ ≈ 130 nm, about 10⁻⁵ at 400 nm. Distance is not what pins the pair. What changes is the oscillation: top right, the momentum content. Ψ_L stays at k_F; Ψ_R collapses to k ≈ 0, and its weight at k_F (the dot) falls by seven decades, in step with the splitting. A smooth end cannot supply that momentum, nor flip the spin that comes with it, so it cannot couple the two halves (Kells et al.; Vuik et al.: nearly opposite spins). Two Majoranas at zero energy, close together, but nothing topological: a quasi-Majorana pair. (If asked: here they are about one to three decay lengths of Ψ_L apart, the “partially separated” case of Vuik et al.; separation alone would give about one decade, not four. Illustrative sweep with the paper’s routine; largest spinor component shown. Press play or drag the slider for questions.)

Same zero-bias peak, no topology: a false positive

The quasi-MZM peak starts before the bulk transition and looks like a true MZM: local spectroscopy cannot tell them apart

Can a different geometry defuse this false positive?

A. Vuik et al. SciPost Phys. 7, 061 (2019) · C.-X. Liu et al. PRB 96, 075161 (2017) · C. Payá et al. PRL 137, 096606 (2026)

Full-shell nanowires through the lens of symmetry

In a full shell, topology is driven by the magnetic flux

The flux \Phi = \pi R^2 B makes the phase of \Delta wind n times: the fluxoid

Away from n\Phi_0, a supercurrent I_\mathrm{s} \propto n - \Phi/\Phi_0 flows; the gap pays for it

At \Phi_0/2, n jumps by one: Little–Parks lobes

Topology only in odd lobes, at B \approx 0.1 T

Zero-bias peaks were seen in the first lobe (Vaitiekėnas et al. 2020), but other mechanisms also produce them

S. Vaitiekėnas et al. Science 367, eaav3392 (2020) · C. Payá et al. PRB 109, 115428 (2024) · M. Valentini et al. Science 373, 82 (2021)

(1:00) The alternative: full-shell nanowires, where the superconductor wraps the whole core. An axial field threads a flux through the shell. The phase of the order parameter must come back to itself around the shell, so it winds an integer number of times, n, the fluxoid: the arrows turn n times going once around. [click] With n fixed, the flux cannot be matched exactly: away from nΦ0 a supercurrent circulates in the shell (the spinning arrows), I_s ∝ n − Φ/Φ0. That current costs kinetic energy, the blue parabola on top, and the superconductor pays for it with a smaller gap, below. [click] At half a flux quantum the next parabola is lower: n jumps by one, the current reverses, vanishes at Φ0 and the gap recovers. Lobe after lobe: the Little–Parks lobes. [click] Topology here is driven by the flux (orbital effect plus radial spin–orbit coupling), at about 0.1 T, and only in odd lobes. Vaitiekėnas and coworkers saw zero-bias peaks in the first lobe in 2020; their origin is debated.

The Hamiltonian splits into angular-momentum sectors

[H, J_z] = 0, \qquad J_z = -i\partial_\varphi + \tfrac{1}{2}\sigma_z + \tfrac{1}{2}n\,\tau_z

H = \bigoplus_{m_J} H_{m_J}, one 2D problem in (r, z) per sector, with

m_J = m_l + \tfrac{1}{2}(\sigma + n\tau) \;\in\; \mathbb{Z} + \tfrac{1 + n}{2}

(m_l \in \mathbb{Z}, \sigma, \tau = \pm 1): m_J = 0 only for odd n: odd lobes

Particle–hole symmetry pairs different sectors: \;\mathcal{P}\,H_{m_J}\,\mathcal{P}^{-1} = -H_{-m_J}

	\mathcal{P} inside the sector	with \mathcal{T}	\mathcal{T} broken
m_J \neq 0	no: maps to -m_J	AI none	A none: trivial
m_J = 0	yes: self-conjugate	BDI \mathbb{Z}	D \mathbb{Z}_2: Majoranas

\mathcal{T} = time reversal × mirror through the axis, \mathcal{T}^2 = +1: the field breaks each but not the product. Exact in the ideal cylinder, broken by disorder.

The m_J = 0 sector turns topological through the orbital effect of the flux, not Zeeman: at B \approx 0.1 T, ~10× lower than in partial shells

P. San-Jose, C. Payá et al. PRB 107, 155423 (2023) · C. Payá et al. PRB 109, 115428 (2024) · A. Altland and M. Zirnbauer, PRB 55, 1142 (1997)

(1:15) In the cylindrical approximation the generalized angular momentum J_z — orbital, spin and the winding of the pairing — is conserved, so H splits into independent sectors, one per m_J. Its eigenvalues are m_l + (σ + nτ)/2: an integer only when the fluxoid n is odd. So the sector m_J = 0 exists only in odd lobes: that is why Majoranas need odd lobes. [click] Particle–hole symmetry, the one we rely on, maps m_J to −m_J: it pairs different sectors. [click] So a sector with m_J ≠ 0 has no particle–hole symmetry of its own: no topological index in 1D. These sectors only host trivial Andreev states, the CdGM analogs. Only the self-conjugate sector m_J = 0 keeps 𝒫, and it is the only one that can host Majoranas. Two columns, because the ideal cylinder has one more symmetry: an effective time reversal, a mirror through the axis combined with time reversal. The field flips under each of them, not under the product, so it survives and squares to +1. With it, the sectors are AI and BDI; without it, once disorder breaks the mirror, A and D. Nothing changes for us: m_J ≠ 0 has no index either way, and of BDI’s integer only the parity survives, the ℤ₂ of class D, like a partial-shell wire. From now on I just say A and D. Contrast with the partial shell, where one class-D Hamiltonian holds all the subbands. [click] And the transition in m_J = 0 is driven by the orbital effect of the flux, not by the Zeeman field: that is why it happens at about 0.1 T, an order of magnitude below the field a partial-shell wire needs to close its gap by Zeeman alone. (If asked: in the sector basis 𝒯 is just complex conjugation, every block is real; (T·M)² = T²M² = (−1)(−1) = +1, hence AI and not AII. It also keeps m_J, unlike the true T.)

Why the transition is inverted in flux

Partial shell: V_\mathrm{Z} grows with B: trivial, then topological above V_\mathrm{Z}^\mathrm{c}

Full shell, m_J = 0: a Majorana nanowire whose Zeeman energy is mostly orbital

V_\mathrm{Z}^\mathrm{eff} = V_\mathrm{Z} + \frac{\tilde\phi}{2}\Bigl(\omega_R + \frac{\alpha}{R_\mathrm{av}}\Bigr), \qquad \tilde\phi = n - \frac{R_\mathrm{av}^2}{R_\mathrm{LP}^2}\frac{\Phi}{\Phi_0}

Topological if V_\mathrm{Z}^\mathrm{eff} > \sqrt{\Gamma_S^2 + \tilde\mu_0^2}. Within a lobe V_\mathrm{Z}^\mathrm{eff} falls with \Phi: MZM at low flux, quasi-MZM at high flux

S. Vaitiekėnas et al. Science 367, eaav3392 (2020) · C. Payá et al. PRB 109, 115428 (2024)

(1:00) You may have noticed that in full shells everything is mirrored: the MZM appears at the lower flux and the quasi-MZM at the higher one. Here is why. In a partial shell the Zeeman energy grows with the field, so the wire goes from trivial to topological at V_Z^c. [click] In a full shell, the m_J = 0 sector is again a Majorana nanowire, but its Zeeman energy is mostly orbital: proportional to the reduced flux φ̃ = n − (R_av/R_LP)² Φ/Φ0, measured from the degeneracy point of the lobe, with ω_R = ħ²/2m*R_av². It is there even at g = 0, and within a lobe it falls as the flux grows. [click] So increasing the flux takes the wire from topological to trivial at Φc: MZM below, quasi-MZM above. Everything that follows in flux looks inverted with respect to the partial shell. (Right: schematic, with R_av²/R_LP² = 0.6 for illustration and the paper’s Φc.)

A full shell is a synthetic vortex that confines Andreev subbands

Abrikosov vortex: Andreev reflection off the pairing wall confines Caroli–de Gennes–Matricon states, \delta E \sim \Delta^2/E_F

Full shell, a synthetic vortex: the winding shell confines Andreev subbands in the core, labelled by m_J, with \delta E \sim \Delta: resolvable

Lobe 0: hard gap. Lobes 1, 2: subgap states that disperse and skew with flux

C. Caroli, P. G. de Gennes, J. Matricon, Phys. Lett. 9, 307 (1964) · P. San-Jose, C. Payá et al. PRB 107, 155423 (2023) · M. T. Deng, C. Payá et al. PRL 134, 206302 (2025)

Majoranas coexist with trivial CdGM analogs at zero energy

m_J \neq 0 (class A): CdGM analogs fill every lobe with trivial subgap states

m_J = 0 (class D): the only sector that can host an MZM, and only in odd lobes

Generically, the Majorana shares zero energy with trivial CdGM analogs: a gapless topological phase

But protected islands exist in parameter space (low \mu), where the MZM keeps a topological minigap (C. Payá et al. PRB 109, 115428)

C. Payá et al. PRB 109, 115428 (2024) · F. Peñaranda et al. PRR 2, 023171 (2020)

The trivial sectors fill first: topology needs \mu > \mu^\mathrm{ts}

C. Payá et al. PRL 137, 096606 (2026)

(1:30) This is the key difference, and it comes from that coexistence. Fill the bands, as happens when walking into the wire from its depleted end. [click] Partial shell (left): all subbands belong to the same class-D Hamiltonian, so whichever fills first can turn topological right above depletion: the ordering of the bands is irrelevant. Full shell (right): computed band bottoms of the first radial mode at k_z = 0 versus m_j = m_l + s_z, one line per spin branch. The radial spin–orbit coupling pushes the minima out to m_j ≈ ±7/2, so the band bottom is not at the m_J = 0 pair (red, m_j = −1/2 on both branches): the class-A CdGM sectors are populated first, and they are trivial. The m_J = 0 pair is split by the flux (an effective Zeeman field for that sector). [click] Topology needs μ inside the window between the two m_J = 0 levels: above the threshold μ^ts, where the lower one fills. (Strictly, the window opens a fraction of a meV higher, as the gap must close.) Along the wire (bottom strip), the stretch where μ(z) < μ^ts is trivial: the trivial skin. (These bands are at ⟨α⟩ = 85 meV nm: the first radial mode only turns topological above ~70 meV nm, unrealistic SOC. At our wire’s 7 meV nm the threshold comes from the second radial mode, ≈ 22 meV, with the whole first mode below it as skin.)

Smooth confinement in full shells

In a grounded shell the gate potential still heals over a length \chi

Gate potential in the core: \nabla^2\phi_g = 0, \;\phi_g = 0 on the grounded shell, \;e\phi_g = \mu_\mathrm{bulk} at the depleted end

e\phi_g(r, z) = \mu_\mathrm{bulk} \sum_{\nu = 1}^{\infty}\frac{2}{k_\nu J_1(k_\nu)}\, J_0\!\Bigl(k_\nu \frac{r}{R}\Bigr)\, e^{-k_\nu z / k_1 \chi}

\chi is a free parameter. Unscreened, the slowest mode gives \chi = R/k_1 \approx 29 nm; screening by the charge that \phi_g displaces needs Schrödinger–Poisson, so we absorb it into \chi, as in partial shells

C. Payá et al. PRL 137, 096606 (2026)

(0:45) The same smooth confinement in a full shell. The charge in the core accumulates at the shell, which we describe with a fixed dome-shaped potential U(r), plus the gate’s φ_g; μ_bulk = U(R) deep in the wire. The tunnel gate adds a correction φ_g: the grounded shell screens it, so it only enters through the uncovered end. Inside, φ_g solves Laplace’s equation (the charge is already in U), vanishes at the shell and depletes the end: a sum of evanescent Bessel modes. The slowest decays over χ = R/k_1, about 29 nm for our wire. What we leave out is the screening of φ_g by the charge it displaces, which would need a self-consistent Schrödinger–Poisson calculation. We absorb it into χ, which becomes the free smoothness parameter, as in partial shells. Projected on the mode that can turn topological, μ(z) heals like the partial-shell profile (right): χ plays the same role in both geometries.

\mu(z) must cross a trivial skin before turning topological

Same construction as in partial shells, now in flux within lobe 1

Inverted in flux: MZM below \Phi^\mathrm{c}, quasi-MZM above (the effective Zeeman energy falls with \Phi within a lobe)

C. Payá et al. PRL 137, 096606 (2026)

Quasi-Majoranas still form in full shells, but deep inside the wire

The quasi-MZM forms and pins to zero, but behind the skin: the probe at the end sees nothing

C. Payá et al. PRL 137, 096606 (2026)

(1:30) The same progression as for the partial shell, now in the full shell, with two views of the m_J = 0 sector versus flux in lobe 1: left, the end LDOS, what a tunnel probe sees; right, the DOS where μ(z) varies, what is inside the wire. χ = 5 nm, a sharp end: both agree, the MZM below Φ^c and the gap reopening above it. [click] χ = 20 nm: inside, nothing has changed; at the end, the peak is already about 80 times weaker. The trivial skin, ≈ 3.5χ wide, now separates the probe from the topological segment. [click] χ = 200 nm: the end is dark. Inside, above Φ^c, the lowest pair starts at zero and splits right away. [click] χ = 500 nm: the pair now stays at zero over a stretch of flux above Φ^c before it splits: the quasi-MZM is growing. [click] χ = 1 µm, paper Fig. 1h: the MZM below Φ^c and the quasi-MZM pinned to zero up to Φ ≈ Φ0, with an exponentially small splitting; at the end, nothing at any flux. [click] Quasi-Majoranas do form in full shells, but behind the skin, where the probe cannot see them. (Illustrative maps with the paper’s wire and routines on the grid of Fig. 1h; the 1 µm DOS is Fig. 1h itself; one colour scale per column.)

The probe sees no quasi-Majorana: no false positives in full shells

C. Payá et al. PRL 137, 096606 (2026)

One length scale both pins the impostor and buries it

Visible: end LDOS above 1 % of its maximum · Energy: end-LDOS splitting and closed-wire pole (partial), m_J = 0 DOS peak (full, this talk)

C. Payá et al. PRL 137, 096606 (2026)

(1:00) Why can’t we escape this? Pinning a quasi-MZM needs a smooth end: smooth on the scale of the Majorana length, so that it cannot couple the two halves of the pair. Partial shell: pinned from χ ≈ 130 nm, and it stays visible at the end for every χ: the two windows overlap, and that overlap is the false positive. [click] Full shell: the same χ also sets the width of the trivial skin, ≈ 3.5χ. Visibility dies at ≈ 19 nm. The hidden pair leaves the gap edge (dotted) at ≈ 35 nm and is pinned below 10⁻² Δ0 from ≈ 55 nm, three times later; by 350 nm it is below 10⁻³ Δ0. (If asked: at Φ^(2) the minigap itself is only ≈ 1.2·10⁻² Δ0, so the pair is born pinned; any threshold gives the same ordering.) [click] There is no window where it is both pinned and visible: the scale that produces the skin is the scale that controls what the probe sees. Quasi-Majoranas cannot be false positives in full shells.

Conclusions

Partial shells

Smooth ends create quasi-MZMs that the probe cannot tell from MZMs

Full shells

Trivial sectors fill first: \mu(z) crosses a trivial skin

No false positives

The same \chi that pins the impostor buries it behind the skin

A zero-bias peak at the end of a full-shell wire is not a quasi-Majorana
No peak? The skin may hide a true MZM: sharp barriers give the most informative measurements

Thank you

Madrid

Instituto de Ciencia de Materiales de Madrid (ICMM), CSIC

Carlos Payá
PhD candidate · defence March 2027

Thesis: full-shell hybrid nanowires. Caroli–de Gennes–Matricon analogs, Majoranas and Josephson junctions

carlosp24.github.io · carlos.paya@csic.es

César Robles
MSc student Pablo San-Jose
ICMM

Supervisor

Elsa Prada
ICMM

Co-supervisor

Ramón Aguado
ICMM

Phys. Rev. Lett. 137, 096606 (2026)
doi.org/10.1103/w9tp-3bbq

Code and data on GitHub

Conclusions

Partial shells

Smooth ends create quasi-MZMs that the probe cannot tell from MZMs

Full shells

Trivial sectors fill first: \mu(z) crosses a trivial skin

No false positives

The same \chi that pins the impostor buries it behind the skin

A zero-bias peak at the end of a full-shell wire is not a quasi-Majorana
No peak? The skin may hide a true MZM: sharp barriers give the most informative measurements

Backup

The conclusion survives disorder and temperature

Short-range disorder below the minigap: MZM peaks survive; quasi-MZM peaks smear out (partial) or never reach the end (full). The same holds for the conductance with all m_J sectors and finite temperature (Suppl. S-IV, S-V).

C. Payá et al. PRL 137, 096606 (2026)

Models and parameters

	Partial shell (1D, Suppl. S-I)	Full shell (Suppl. S-II)
Common	m^* = 0.023\,m_e, \Delta_0 = 0.23 meV	m^* = 0.023\,m_e, \Delta_0 = 0.23 meV
Filling, SOC	\mu_\mathrm{bulk} = 2 meV, \alpha = 40 meV nm	\mu_\mathrm{bulk} = 22.8 meV, \Delta U = 60 meV, \langle\alpha\rangle = 7 meV nm
Coupling	\Gamma_\mathrm{NS} = 3\Delta_0, V_C = 2V_\mathrm{Z}^\mathrm{c}	\Gamma_\mathrm{NS} = 40\Delta_0, g = 10
Geometry	—	R = 70 nm, d = 10 nm, \xi_d = 70 nm
Smoothness (Fig. 1)	\chi = 200 nm	\chi = 1 µm
Disorder (Fig. 3)	A_\mathrm{imp}/\sqrt{N} = 0.32\Delta_0	A_\mathrm{imp}/\sqrt{N} = 0.02\Delta_0

Full shell: solid InAs-like core with a dome-shaped potential U(r), radial Rashba SOC, flux and Zeeman; the Al shell is integrated out into a Little–Parks (Abrikosov–Gor’kov) self-energy; solved per m_J sector.

C. Payá et al. PRL 137, 096606 (2026)

Electrostatics and skin widths

Projected on the m_J = 0, m_r = 1 mode that turns topological:

\mu(z) \simeq \mu_\mathrm{bulk}\bigl(1 - 0.85\, e^{-z/\chi}\bigr)

Case	Local phases along the wire
Partial, V_\mathrm{Z}^{(2)}	topological for z < 1.86\chi, trivial beyond
Full, \Phi^{(1)}	skin for z < 3.42\chi, then topological
Full, \Phi^{(2)}	skin for z < 3.50\chi, topological up to 7.01\chi

Full-shell end signal below 1 % of its maximum for \chi > 19 nm (MZM and quasi-MZM)

Phase diagram vs \mu and \langle\alpha\rangle at \Phi = \Phi_0/2, with the thresholds of the m_r = 0 and m_r = 1 modes (paper Fig. 4c)

C. Payá et al. PRL 137, 096606 (2026)

Origin of the skin: radial modes m_r = 0 and 1

C. Payá et al. PRL 137, 096606 (2026)

Conductance with all m_J sectors, and temperature

C. Payá et al. PRL 137, 096606 (2026)

Reproduce it: the partial-shell wire in Quantica.jl

lat = LP.square(; a0) |> supercell((1, 0))         # 1D chain, a0 = 5 nm

p2 = @onsite((r; μ = μ) -> σ0τz * (2t - μ)) +       # kinetic energy
     hopping((r, dr) -> -t * σ0τz, range = a0)

rashba = @hopping((r, dr; α = α) ->                  # Rashba SOC
     α * im * dr[1] / (2a0^2) * σyτz, range = a0)
zeeman = @onsite((; B = B) -> σzτ0 * B)              # Zeeman

hSM = lat |> hamiltonian(p2 + rashba + zeeman; orbitals = Val(4))
Σ! = @onsite!((o; ω = 0, B = B) -> o + τΓ * ΣSC(Δ0 * Δf(B; Bc), ω))
hSC = hSM |> Σ!                                      # proximity self-energy

h, L = build_barrier(hSC, pw, χ)                     # smooth end, μ(z)
g = h |> attach(gSC; region = r -> r[1] == L) |> greenfunction()

src/builders/apartial.jl and barrier.jl in the paper’s repository; the full shell uses FullShell.build_cyl